1MA1 · S3–S4StatisticsHigherGrades 6–8

Cumulative Frequency & Box Plots

Q: Why use the IQR instead of the range?

The IQR ignores the most extreme quarter at each end, so it is not distorted by outliers.

Q: Does the cumulative frequency graph start at zero?

Yes — plot zero at the lower boundary of the first class.

Q: What does a longer box mean?

A larger IQR: the middle half of the data is more spread out.

For grouped data you cannot find the exact median, but a $\textbf{cumulative frequency graph}$ lets you estimate it — along with the quartiles, the interquartile range and any percentile. Those five key values then summarise the data in a $\textbf{box plot}$ , the clearest way to compare two distributions side by side.

50 min Statistics

The big picture

Cumulative frequency is a running total: how many values are less than or equal to each upper class boundary. Plotting it produces an S-shaped curve, and reading across from $\tfrac{n}{2}$ , $\tfrac{n}{4}$ and $\tfrac{3n}{4}$ gives the median and quartiles.

The interquartile range measures the spread of the middle half of the data, so it ignores extreme values — which is why Edexcel prefers it to the range for comparisons. A full comparison states which group has the higher median and which has the smaller IQR, each in context.

What you'll learn

Complete cumulative frequency tables and draw cumulative frequency graphs
Estimate the median, quartiles, IQR and percentiles from a graph
Draw and interpret box plots
Compare distributions using the median and interquartile range

Cumulative frequency tables and graphs

Add frequencies as you go down the table to get the running total. The final cumulative frequency equals the total number of values.

Plot each cumulative frequency at the $\textbf{upper class boundary}$ of its class, starting from zero at the lower boundary of the first class.

Join the points with a smooth curve or straight lines.

Example 1Times (minutes) for 80 runners:

20 \le t < 25

: 6;

25 \le t < 30

: 18;

30 \le t < 35

: 30;

35 \le t < 40

: 20;

40 \le t < 45

: 6. Write the cumulative frequencies and the points to plot.

1Running totals: 6, 24, 54, 74, 80.

2Plot

(25, 6)

(30, 24)

(35, 54)

(40, 74)

(45, 80)

, starting at

(20, 0)

AnswerCumulative frequencies 6, 24, 54, 74, 80 plotted at 25, 30, 35, 40, 45.

Your turnSame method, new numbers — try it before moving on.

Class frequencies are 5, 12, 20, 9 and 4. What is the final cumulative frequency?

Your answer

Show worked solution

Solution

1Running totals: 5, 17, 37, 46, 50.

Answer: $50$

Tip — Plotting at midpoints is the classic cumulative frequency error. The running total counts everything up to the end of the class, so plot at the upper boundary.

Reading the graph

Median: read across from $\tfrac{n}{2}$ on the cumulative frequency axis to the curve, then down to the value.

Lower quartile $Q_{1}$ at $\tfrac{n}{4}$ ; upper quartile $Q_{3}$ at $\tfrac{3n}{4}$ . Interquartile range $=$ $Q_{3} - Q_{1}$ .

To find how many values are above or below a given value, read up from that value and across to the cumulative frequency.

\text{IQR} = Q_{3} - Q_{1}

Interquartile range.

Example 1For the 80 runners, the curve gives

Q_{1} = 29.5

, median

= 33

and

Q_{3} = 36.5

minutes. Find the IQR and explain what it tells you.

1IQR

= 36.5 - 29.5

AnswerIQR

= 7

minutes: the middle half of the runners finished within a 7-minute window.

Your turnSame method, new numbers — try it before moving on.

A data set has lower quartile 18 and upper quartile 31. Find the interquartile range.

Your answer

Show worked solution

Solution

1 $31 - 18 = 13$ .

Answer: $13$

Why this works

Reading values from a curve is an estimate, because the curve assumes values are spread smoothly within each class. Mark schemes accept a small range of readings.

Box plots

A box plot shows five values on a scale: minimum, $Q_{1}$ , median, $Q_{3}$ and maximum.

The box runs from $Q_{1}$ to $Q_{3}$ with a line at the median; the whiskers extend to the minimum and maximum.

Box plots drawn on the same scale make comparisons immediate.

Your turnSame method, new numbers — try it before moving on.

A box plot has minimum 12 and maximum 58. What is the range?

Your answer

Show worked solution

Solution

1 $58 - 12 = 46$ .

Answer: $46$

Tip — Draw box plots accurately against a scale. A box plot without a scale cannot be read.

Comparing distributions

Compare the $\textbf{medians}$ to say which group is higher on average, and the $\textbf{IQRs}$ (or ranges) to say which is more consistent.

Write each comparison in context, with the values.

Example 1Box plots of reaction times: Group A median 0.42 s, IQR 0.10 s; Group B median 0.36 s, IQR 0.18 s. Compare the groups.

1Median: B is lower, so B reacted faster on average.

2IQR: A is smaller, so A’s times were more consistent.

AnswerGroup B reacted faster on average (median 0.36 s vs 0.42 s), but Group A was more consistent (IQR 0.10 s vs 0.18 s).

Think like an examiner

Plot cumulative frequencies at upper class boundaries.
Show reading lines on the graph for medians and quartiles.
Compare distributions using both an average and a spread, in context.

Cumulative frequency

\text{median at } \tfrac{n}{2}

Median.

Q_{1} \text{ at } \tfrac{n}{4},\ Q_{3} \text{ at } \tfrac{3n}{4}

Quartiles.

Q_{3} - Q_{1}

Interquartile range.

Watch out for these

Plotting cumulative frequency at midpoints.

Plot at upper class boundaries.

Reading the median from the frequency axis value

n

Read across from

\tfrac{n}{2}

Comparing only medians.

Also compare spread with the IQR.

Saying the higher median group is "better" without context.

Interpret: faster, taller, higher scores, and so on.

Stretch yourself

Using the 80 runners’ data, estimate how many runners took longer than 38 minutes, assuming values are spread evenly within classes.

Hint — Find how many finished in under 38 minutes by working within the $35 \le t < 40$ class.

Reveal solution

SolutionBy 35 minutes, 54 runners had finished. The

35 \le t < 40

class holds 20 runners over 5 minutes; spreading them evenly,

\tfrac35 \times 20 = 12

finished between 35 and 38 minutes. So about

54 + 12 = 66

finished within 38 minutes, and

80 - 66 = 14

took longer.

\blacksquare

Questions students ask

Why use the IQR instead of the range?

The IQR ignores the most extreme quarter at each end, so it is not distorted by outliers.

Does the cumulative frequency graph start at zero?

Yes — plot zero at the lower boundary of the first class.

What does a longer box mean?

A larger IQR: the middle half of the data is more spread out.

Key takeaways

Cumulative frequency is a running total, plotted at upper class boundaries.
Median at $\tfrac{n}{2}$ , quartiles at $\tfrac{n}{4}$ and $\tfrac{3n}{4}$ .
IQR $= Q_{3} - Q_{1}$ measures the spread of the middle half.
Box plots show minimum, $Q_{1}$ , median, $Q_{3}$ and maximum.
Compare medians and IQRs, in context.

Exam-style questions

Practise it the way it is examined

3 original questions · 6 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 62 marks

Cumulative frequencies for 100 people: $\le 10$ : 8, $\le 20$ : 30, $\le 30$ : 65, $\le 40$ : 90, $\le 50$ : 100.

(a)How many people had a value greater than 40?[1]
Show mark scheme (a)
- B1 $10$
Answer: $10$
(b)Which class contains the median?[1]
Show mark scheme (b)
- B1 $20$ – $30$

Question 2Grade 72 marks

A box plot shows minimum 15, lower quartile 22, median 30, upper quartile 41, maximum 55.

(a)Work out the interquartile range.[1]
Show mark scheme (a)
- B1 $19$
Answer: $19$
(b)What percentage of the data is greater than 41?[1]
Show mark scheme (b)
- B1 $25\%$
Answer: $25\%$

Question 3Grade 82 marks

Group A has median 24 and IQR 8. Group B has median 21 and IQR 15. Compare the two groups.[2]
Show mark scheme
- B1Group A has a higher average (median)
- B1Group A is more consistent (smaller IQR)

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

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